Fabrici, Igor

English

Fabrici Igor

University of Szeged

On invertible elements in compact semigroups: By I. FABRICI in Bratislava (CSSR) . The paper [6] investigates the structure of abstract semigroups containing invertible elements. The purpose of this paper is to study the structure of compact semigroups containing invertible elements. Throughout this paper S will denote a Hausdorff compact semigroup. In the first place we set down some notions and statements. An element a of S is called totally maximal, if SaS=S. An element a of S is called left (right, two-sided) invertible if Sa = S (aS= S, aSa = S). Further, Jf is the set of elements of a semigroup S which are neither left nor right invertible, is the set of elements of S which are left invertible but not right invertible, 3t is the set of elements of 5 which are right invertible but not left in-vertible, and finally ^ is the set of elements of S which are both left and right invertible. From [5] it is known that each of the sets &t, <& is a subsemi-group of the semigroup S. Denote by L* the maximal proper left ideal of a semigroup S, which contains every proper left ideal of S. The maximal proper right ideal R* and the maximal proper two-sided ideal M* are defined similarly. All ideals in this paper are considered in the algebraic sense. We admit in our considerations that the ideals L*, R*, M* possibly are void sets. L e m m a 1. [6] If in a semigroup S there exists at least one left invertible element, then S contains the unique maximal proper left ideal L* and the complement of this ideal is the set of left invertible elements of S; hence S = L* U i f U Remark 1. A similar statement holds if S contains at least one right invertible element. L e m m a 2. [2] Let S be a compact semigroup. If the ideal L*(R*) exists in S, then M* and M* =L* (M*=R*) also exist in S. From [6] it is known that every left invertible or right invertible element of a semigroup S is totally maximal. The converse statement does not hold. 272 I. Fabrici Let Ji denote the set of all totally maximal elements of S which are neither left nor right invertible. Then, evidently J /Q C/C. C o r o l l a r y . Let S be a compact semigroup, which contains invertible elements. Then every totally maximal element is either left invertible or right invertible. T h e o r e m la. Let S be a compact semigroup. Then P = is a closed subset of the compact semigroup S. Proof. Since P = {aZS: Sa = S}, there exists for an arbitrary b£S an x£S such that xa = b. In order to prove that P is closed it is sufficient to show that for an arbitrary a^P(P means the closure of P) and for an arbitrary ¿ £ S there exists an x£S such that the relation xa = b holds, since in this way it will be proved that P = P. Let us assume that this is not true. This means that for some b £ S the equation xa — b has no solution in S and, therefore xa^b for every x£S. Since S is a Haus-dorff space, it follows from the continuity of multiplication that there exist neigh-bourhoods o(x) £ O(x), ox(a) £ 0(a) and ox(b) £ 0(b) such that ox(b)f][o(x), o » ] = 0 , where 0(x), 0(a) and 0(b) are complete systems of neighbourhoods of the elements x, a, b. Let us consider a system of neighbourhoods {o(x)}, JC £ S. It is evident that S = U o(x). Since S is a compact semigroup, there exists such a finite system xiS o^Vj), o(x2), ...,o(xn), which also covers S. For /' = 1,2, ...,n we have 0xi(b)C\[o(x^)- oxi(a)] = 0. n Evidently, there exist such neighbourhoods o(b) £ 0(b), o(a) 6 0(a), that o(b) c f | oxi(b) i— 1 n and o(a)d f) oxi(a). But then we have o(b)r\[o(xi)-o(a)\ = <A, n for / = 1,2, Since S = U o(xf), it follows from the preceeding relations: i=t ( * ) O ( £ ) N [ S - O ( A ) ] = 0 . We show that the last relation is not correct. Since a6P, it follows that in every neighbourhood of a there exists at least one element of P. Therefore there exists such an element that (€o(a) and, since there exists an such that b = But ¿>€o(6), so b = tj-l;£S-o(d), and this contradicts relation ( * ) . Analogously, one proves the following T h e o r e m lb. Let S be a compact semigroup. Then Q— ffl U^ is a closed subset Qf S. Invertible elements in compact semigroups 273-C o r o l l a r y . P and Q are compact subsets of S. We say that a semigroup S is left simple (right simple, simple) if S contains no proper left (right, two-sided) ideal of S, distinct of <S and the void set. T h e o r e m 2a. Let S be a compact semigroup. Then P = (J $ is a left simple compact subsemigroup of S. Proof. Let a, be P. Then Sa = S, Sb = S, S(ab) = (Sa)b = Sb = S. This means that ab £ P and P is a subsemigroup of S. Assume that L' cr S — L* is a proper left ideal of S-L*. Then Lemma 2 implies: S(L'UL*) = SL'U SL* = (S-L*)L'U U L*L' U SL* c L' U L* and this is a contradiction with the assumption that L* contains every proper left ideal of S. The proof of the following statement is analogous. T h e o r e m 2b. Let S be a compact semigroup. Then is a right simple compact subsemigroup of S. : Let e be an idempotent of a compact semigroup S. We say that an element a 6 S belongs to the idempotent e, if e is the unique idempotent of the closure A of the semigroup A = {a, a 2, ...}. Let us denote by Kx the set of all elements of a semigroup S, which belongs to the idempotent ex; we shall call it a AT-class. From [1] it is known that any compact semigroup S can be written as the union of disjoint A^-classes. We say that a group Gx is a maximal group belonging to the idempotent ex, if Gx contains ex and if there exists no group G'T^G, such that C . c C ' c J i , . T h e o r e m 3a. Either of the subsemigroups L*, P of a compact semigroup S is the union of some K-classes of S. Proof. To. prove our statement it is sufficient to show that no A'-class can have a non-void intersection with both subsemigroups. Let us assume that KXC\L*9±Q and KXC\P?±® holds for some /C-class Kx. Let a£KxDL*. a£Kx means that a belongs to the idempotent ex. But also a£L*. Let us consider the principal left ideal generated by a: (a)L = aU Sa. (a)LcL* and (a)L is a closed subsemigroup of a compact semigroup S, therefore (a)L is also a compact subsemigroup. So we have A Q(a)L where A = {a,a 2, ...}.The sub-semigroup A contains the unique idempotent and since a £KX this idempotent must be ex. We have obtained that ex£Acz (a)L cz L*. Let now b£Kxf]P. The element b also belongs to the idempotent ex. But since b£P and P is a compact subsemi-group of the semigroup 5, we have ex£BczP for B = {b,b 2, ...}. We have ob-tained that ex € L* and at the same time ex € P. But this is impossible, since L* Pi P = 0. 8 A 274 I. Fabrici Analogously, we can prove T h e o r e m 3b. Either of the subsemigroups R*, Q of a compact semigroup S is the union of some K-classes of S. Remark 2. In [3] it is proved that a left (right) simple semigroup having at least one idempotent is a disjoint union of algebraically isomorphic groups C4. But P is a left simple subsemigroup, Q is a right simple subsemigroup. Moreover, either of P, Q is a compact subsemigroup, so they contain at least one idempotent. From this we have T h e o r e m 4. Either of the subsemigroups P, Q of a compact semigroup S is a disjoint union of topologically isomorphic maximal groups G x . Proof. The statement that P(Q) is a disjoint union of maximal algebraically isomorphic groups Ga follows from Remark 2 and Theorems 3a and 3b. It is only necessary to show that these groups are isomorphic also topologically. It is known from [3] that if e„ 6 P then Gx = etP and for x 6 Ga (1) x-efix is an isomorphic mapping of the group Gx onto Gp and for y£Gf (2) . is the inverse mapping.. But according to the assumption, the multiplication in S is continuous. This means that the transformation (1) is a continuous transformation of a topological group Ga onto the topological group Gfi, and the inverse transformation (2) is a continuous transformation of Ge onto Gx. Hence, Ga and Gp are topologically isomorphic. C o r o l l a r y . If in a compact semigroup S, Sg^0 (3%?^%) then the semigroup jSf (31) contains at least one idempotent. T h e o r e m 5a. If a compact semigroup S contains at least one left invertible element, then S contains at least one right unit. Proof. From Theorem 2a. we know that P contains at least one idempotent. Let e1€i> . Then, evidently, Sex = S. Let x£S be an arbitrary element. Sei = S implies yex = x for some y£S. Hence, xe1 — (ye1)e1=ye\=ye1 = x. This means that ex is a right unit of 5. Analogously we can prove T h e o r e m 5b. If a compact semigroup S contains at least one right invertible element, then S contains at least one left unit. T h e o r e m 6. Let S be a compact semigroup. Then only one of and can be non-empty. Invertible elements in compact semigroups 275 P r o o f . Let . 2 У 0 , F r o m [5] w e k n o w that <8 is a s u b g r o u p o f S and its unit is the unit o f 5 , and Corol lary o f T h e o r e m 4 impl ies that i f conta ins at least o n e i d e m p o t e n t which is a right unit o f S, this i s a contradic t ion . C o r o l l a r y 1. If S is a compact semigroup, then only the following cases are possible: 1) S = JT, 2) S=<£, 3) S = 4 ) S = 5 S = J f U i f , 6) S = / U « , and!) S=X\}9. T h e o r e m s l a , l b , and 6 imply C o r o l l a r y 2. The subsemigroups i f , 2k, and H of a compact semigroup S are compact subsemigroups. T h e o r e m s 2a, 2b, 4 and 6 imply C o r o l l a r y 3. The subsemigroup i f is left simple (right simple) and both are disjoint unions of topologically isomorphic maximal groups References [1] Ш. Шварц, К теории хаусдорфовых полугрупп, Чехослов. мат. журнал, 5 (1955), 1—23. [2] Ш. Шварц, О топологических полугруппах с односторонними единицами, Чехослов. мат. журнал, 5 (1955), 153—162. [3] Ш. Шварц, Структура простых полугрупп без нуля, Чехослов. мат. журнал, 1 (1951), 51—65. [4] Ш. Шварц, Об увеличительных элементах в теории полугрупп, Доклады Акад. Наук СССР, 4 (1955), 697—698. [5] Е. С. Ляпин, Полугруппы (Москва, 1960). [6] И. Фабрици, Об обратимых элементах полугруппы и их отношении к увеличительным элементам полугруппы, Mal.-Iyz. casopis, 15 (1965), 177—185. [7] I. FABRICI, О uplne maximalnych prvkoch v pologrupach, Mat.-fyz. casopis, 13 (1963), 16—19. [8] A . H . CLIFFORD—G. B . PRESTON, The algebraic theory of semigroups. I (Providence, 1961). (Received Aug. 28, ¡968) 

On invertible elements in compact semigroups

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On invertible elements in compact semigroups

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